Aspects of Reflection Groups

April 11, 12, 13

The students attending the reflection groups course did projects on topics related to finite Coxeter groups, root systems and their applications. Here are the details of the mini conference and the project reports.

Date Time Place Speaker Title
April 11 3:30 - 4:05 pm LH 6 Shraddha Specht method for constructing irreducible representations
of groups of type $A_n$ and $B_n$.
April 11 4:10 - 4:40 pm LH 6 Kavita Classification of quivers.
April 12 10:30 - 11:05 am LH 6 Rijul Symmetries of some regular polytopes.
April 12 11:10 - 11:40 am LH 6 Shreejit Construction and symmetries of Icosahedron.
April 13 10:30 - 11:05 am Seminar Dipankar An introduction to triangle groups.
April 13 11:10 - 11:40 am Seminar Ayan M. The three reflections theorem.
April 13 11:50 - 12:25 pm Seminar Sayantan Permutahedra and matrix mutation.
April 13 12:30 - 1:05 pm Seminar Apurv The topological aspects of braid groups.
April 13 2:15 - 2:50 pm Seminar Ayan S Reflections on manifolds.
April 13 2:55 - 3:30 pm Seminar Sarjick Classification of semisimple Lie algebras.
April 13 3:35 - 4:05 pm Seminar Sudipta An introduction to Weyl groupoids.

The Abstracts

Specht method for constructing irreducible representations of groups of type $A_n$ and $B_n$: Representation theory of finite groups is an important branch of mathematics. In the context of reflection groups one can see a rich interplay between algebra and combinatorics. In this talk I shall focus on 2 classes of reflection groups; the symmetric groups (type $A$) and the octahedral groups (type $B$). I will explain, in detail, the Specht method of constructing all irreducible representations of these groups upto equivalence.

Classification of quivers: An integral quadratic form (called the 'Euler form') defined on directed graphs (quivers) is intimately connected to the representation theory of some finite-dimensional algebras. In my talk I will describe this connection and the classification of quivers induced by the Euler form.

Symmetries of some regular polytopes: The study of regular polytopes has a long history in mathematics. A seminal theorem of Coxeter says that symmetry groups of such polytopes can be realized as reflection groups. In this talk I shall describe some these polytopes and their symmetry groups. Time permitting, I will also outline their classification.

Construction and symmetries of Icosahedron: The icosahedron is one of the important platonic solids in Euclidean geometry, admitting a highly rich group of symmetries. In this talk I shall give an outline of Taylor's method of constructing an Icosahedron. A brief sketch of Kepler's method will also be given and then I shall establish that the polytope we construct is unique up to an isometry. In the end, some nice geometrical properties of the icosahedron and a few interesting facts about its symmetry group will be discussed, with special emphasis on the orientation-preserving or rotational symmetries.

An introduction to triangle groups: A triangle group is a (possibly) infinite reflection group. It is realized, geometrically, by sequences of reflection across the sides of a triangle. There are three types of
triangle groups; Euclidean,Spherical and Hyperbolic. In this talk I will describe the Euclidean and Spherical triangle groups. Time permitting, I will also talk about hyperbolic triangle groups.

The three reflections theorem: It is well known that an isometry of an $n$-dimensional Euclidean space is a product of at most $n+1$ reflections. Aim of my talk is to show that a similar statement is true in the context of hyperbolic plane. I will explain the upper-half plane model and the required concepts from hyperbolic geometry. Finally, conclude that a hyperbolic isometry is a product of at most $3$ hyperbolic reflections.

Permutahedra and matrix mutations: For a reflection group W, the associated W-permutahedron is the convex hull of the W-orbit of a generic point. In this talk, I shall first describe properties of a W-permutahedron associated to classical root systems. Later, I shall explain the operations of the matrix mutation using concrete examples.

Topological aspects of braid groups: Artin groups is a class of groups naturally associated to reflection groups. In this talk, I will first describe their construction. Then I will concentrate on a well known class; the braid groups. These groups are associated to symmetric groups. They are also of independent interest because of their connection to configuration spaces. I will state this connection and if time permits I will also skectch the proof, by Arnold, that these are aspherical spaces.

Reflections on manifolds: Reflection on a smooth manifold is an involutive self-diffeomorphism whose fixed poit set is a codimension-1 submanifold. A discrete subgroup of diffeomorphisms generated by finitely many reflections is called as a manifold reflection group. The situation is very similar that of a reflection group acting on a finite-dimensional vector space. In this talk I will first describe an analouge of reflection arrangements and then sketch the proof of the fact that these groups have a Coxeter presentation.

Classification of semisimple Lie algebras: It was shown in the class that the finite reflection groups are classified using the Coxeter diagrams. In this talk I will describe how semisimple Lie algebras are also classified using such diagrams. I will describe, first, how to associate a Weyl group to a semisimple Lie algebra and then how to use the classification theorem done in class.

An introduction to Weyl groupoids: The study of root systems and Weyl groups naturally arise in study of Lie algebras.In the last decades,the concept of a lie algebra has been generalized in many different directions.Recent results on pointed Hopf algebras have led to yet another symmetry structure,the Weyl groupoid. In my talk , I will give an axiomatic definition of Weyl groupoids and talk about connections with crystallographic arrangements.